Philosophy of Mathematics, Misc

By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long time, (...) the real, ideal numbers of Plato’s Ideal mathematics eliminates many mathematical problems, extends the capabilities of modelling, and improves mathematics. (shrink)

Hermann Grassmann’s 1861 work [2] was probably the first attempt at an axiomatic approach to arithmetic. The historical significance of this work is enormous, even though the set of axioms has proven to be incomplete. Basing on the interpretation of Grassmann’s theory provided by Hao Wang in [4], I present its detailed discussion, define the class of models of Grassmann’s arithmetic and discuss a certain axiom system for integers, modeled on Grassmann’s theory. At the end I propose to modify the (...) set of axioms of Grassmann’s arithmetic, which consists in adding an elementary sentence and removing a non-elementary one. I prove that after this modification the only model of the theory up to isomorphism is the standard model. (shrink)

It is often said that ‘every logical truth is obvious’ (Quine 1970: 82), that the ‘axioms and rules of logic are true in an obvious way’ (Murawski 2014: 87), or that ‘logic is a theory of the obvious’ (Sher 1999: 207). In this chapter, I set out to test empirically how the idea that logic is obvious is reflected in the scholarly work of logicians and philosophers of logic. My approach is data-driven. That is to say, I propose that systematically (...) searching for patterns of usage in databases of scholarly works, such as JSTOR, can provide new insights into the ways in which the idea that logic is obvious is reflected in logical and philosophical practice, i.e., in the arguments that logicians and philosophers of logic actually make in their published work. (shrink)

This article introduces a method for programming orbitals at the Planck level. Mathematical probability orbitals are replaced with units of `orbit momentum' with orbital regions derived from geometrical imperatives rather than abstract forces. In this approach the electron does not orbit around a nucleus but rather is maintained within an orbital region by the confines of the geometry of the orbital (this orbit momentum is the orbital). There is no electron transition between orbitals, rather the existing orbital is exchanged for (...) the new orbital by the momentum of the incoming photon. The Rydberg formula describes this 2-stage process of wave addition followed by wave subtraction. As the electron is physically linked to the nucleus by the orbital an electric force is not required, instead a charged momentum (the sqrt of momentum) is presumed. A gravitational orbit is the sum of individual gravitational orbitals as physical links of orbit momentum, a gravitational force is not required. Mass is replaced by units of Planck mass, with the orbitals linking these Planck mass units. Thus the moon is not orbiting the earth, instead it is propelled by these orbital momenta, its path the sum of this momentum. As orbitals have different momentum densities, movement between orbitals requires a change in momentum, an orbital (momentum) buoyancy. Nuclear binding energy, ionization energy and escape velocity are measures of the momentum required to completely erase the orbitals. (shrink)

The Simulation Hypothesis proposes that all of reality is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. Outlined here is a method for reproducing relativistic mass, space and time from the Planck level. For a virtual universe the model uses a 4-axis hyper-sphere that expands in incremental steps (the simulation clock-rate). Virtual particles that oscillate between an electric wave-state and a mass point-state are mapped within this hyper-sphere, the oscillation driven (...) by this expansion. Particles are assigned an N-S axis which determines the direction in which they are pulled along by the expansion, thus an independent particle momentum is not required. Only in the mass point-state do particles have fixed hyper-sphere co-ordinates. The rate of expansion translates to the speed of light and so in terms of the hyper-sphere co-ordinates all particles (and objects) travel at the speed of light, time (as the clock-rate) and velocity (as the rate of expansion) are therefore constant, however photons, as the means of information exchange, are restricted to lateral movement across the hyper-sphere thus giving the appearance of a 3-D space. Lorentz formulas are used to translate between this `3-D space' and the hyper-sphere, relativity as the mathematics of perspective. (shrink)

The simulation hypothesis proposes that all of reality is an artificial simulation. In this article I describe a simulation model that derives Planck level units as geometrical forms from a virtual (dimensionless) electron formula $f_e$ that is constructed from 2 unit-less mathematical constants; the fine structure constant $\alpha$ and $\Omega$ = 2.00713494... ($f_e = 4\pi^2r^3, r = 2^6 3 \pi^2 \alpha \Omega^5$). The mass, space, time, charge units are embedded in $f_e$ according to these ratio; ${M^9T^{11}/L^{15}} = (AL)^3/T$ (units = (...) 1), giving mass M=1, time T=$2\pi$, length L=$2\pi^2\Omega^2$, ampere A = $(4\pi \Omega)^3/\alpha$. We can thus for example create as much mass M as we wish but with the proviso that we create an equivalent space L and time T to balance the above. The 5 SI units $kg, m, s, A, K$ are derived from a single unit u = sqrt(velocity/mass) that also defines the relationships between the SI units; kg = $u^{15}$, m = $u^{-13}$, s = $u^{-30}$, A = $u^{3}$, $k_B = u^{29}$. To convert MLTA from the above $\alpha, \Omega$ geometries to their respective SI Planck unit numerical values (and thus solve the dimensioned physical constants $G, h, e, c, m_e, k_B$) requires an additional 2 unit-dependent scalars. Results are consistent with CODATA 2014. The rationale for the virtual electron was derived using the sqrt of momentum P and a black-hole electron model as a function of magnetic-monopoles AL (ampere-meters) and time T. (shrink)

This volume collects nine essays that investigate the work of Gottlob Frege. The contributors address Frege’s work in relation to literature and fiction (Dichtung), the humanities (Geisteswissenschaften), and science (Wissenschaft). Overall, the essays consider internal connections between different aspects of Frege’s work while acknowledging the importance of its philosophical context. -/- There are also further common strands between the papers, such as the relation between Frege’s and Wittgenstein’s approaches to philosophical investigations, the relation between Frege and Kant, and the place (...) of Frege’s work in the philosophical landscape more generally. The volume is therefore of direct relevance to several current debates in philosophy in general, in addition to Frege and Wittgenstein research in particular. -/- Even though Frege’s great significance for contemporary philosophy is not disputed, the question of how we are to understand the character and aims of his project is debated. The debate has a starting point in Frege’s specific conception of logic. The volume elucidates this conception as well as the relation between natural language and the Begriffsschrift. It will help philosophers, researchers, and students better understand the nuances of this great thinker. By extension, it will also help readers seeking to understand Wittgenstein’s approach to philosophical difficulties and his struggle to find an apt form of presentation for his philosophical investigations. (shrink)

Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes on to characterize this (...) reality as an ‘ideal reality’ that ‘governs’ the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by a what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems. (shrink)

Two seemingly contradictory tendencies have accompanied the development of the natural sciences in the past 150 years. On the one hand, the natural sciences have been instrumental in effecting a thoroughgoing transformation of social structures and have made a permanent impact on the conceptual world of human beings. This historical period has, on the other hand, also brought to light the merely hypothetical validity of scientific knowledge. As late as the middle of the 19th century the truth-pathos in the natural (...) sciences was still unbroken. Yet in the succeeding years these claims to certain knowledge underwent a fundamental crisis. For scientists today, of course, the fact that their knowledge can possess only relative validity is a matter of self-evidence. The present analysis investigates the early phase of this fundamental change in the concept of science through an examination of Hermann von Helmholtz's conception of science and his mechanistic interpretation of nature. Helmholtz (1821-1894) was one of the most important natural scientists in Germany. The development of this thoughts offers an impressive but, until now, relatively little considered report from the field of the experimental sciences chronicling the erosion of certainty. (shrink)

For over half a century I have been interested in the role of intuitive spatial reasoning in mathematics. My Oxford DPhil Thesis (1962) was an attempt to defend Kant's philosophy of mathematics, especially his claim that mathematical proofs extend our knowledge (so the knowledge is "synthetic", not "analytic") and that the discoveries are not empirical, or contingent, but are in an important sense "a priori" (which does not imply "innate") and also necessarily true. -/- I had made my views clear (...) in courses on philosophy of science and mathematics when teaching at Sussex University (from 1964) which was why one of our former students, Mary Pardoe (then Mary Ensor) who had become a mathematics teacher informed me, while visiting the university, that she had found a new diagrammatic proof of the triangle sum theorem. I reported her proof in some papers and presentations on methods of representation and reasoning, e.g. here, but neither she nor I has encountered anyone else who knew about the proof. (shrink)

This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.

Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic: Alfred Tarski and other Polish logicians such as Adolf Lindenbaum developed a general theory of logics at the end of the 1920s based on consequence operations and logical matrices. The subject was revived after the flowering of thousands of new logics during the last (...) thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity. This book contains recent works on universal logic by first-class researchers from all around the world. The book is full of new and challenging ideas that will guide the future of this exciting subject. It will be of interest for people who want to better understand what logic is. Tools and concepts are provided here for those who want to study classes of already existing logics or want to design and build new ones. (shrink)

Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is a (...) point about set theory, not number theory. (shrink)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

Sebastian Grève interviews Felix Mühlhölzer on his work on the philosophy of mathematics.

Dynamical systems are mathematical objects meant to formally capture the evolution of deterministic systems. Although no topological constraint is usually imposed on their state spaces, there is prima facie evidence that the topological properties of dynamical systems might naturally depend on their dynamical features. This paper aims to prepare the grounds for a systematic investigation of such dependence, by exploring how the underlying dynamics might naturally induce a corresponding topology.

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

While the appeal of scientific materialism has been weakened by developments in theoretical physics, chemistry and biology, Pythagoreanism still attracts the allegiance of leading scientists and mathematicians. It is this doctrine that process philosophers must confront if they are to successfully defend their metaphysics. Peirce, Bergson and Whitehead were acutely aware of the challenge of Pythagoreanism, and attempted to circumvent it. The problem addressed by each of these thinkers was how to account for the success of mathematical physics if the (...) world consists of creative processes. In this paper I critically examine the nature of the challenge posed by Pythagoreanism to process philosophy and examine the efforts by process philosophers, particularly Whitehead, to overcome it, and offer some suggestions for advancing these efforts. (shrink)

The generality relativist has been accused of holding a self-defeating thesis. Kit Fine proposed a modal version of generality relativism that tries to resist this claim. We discuss his proposal and argue that one of its formulations is self-defeating.

Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving (...) process? What is the ontological status of a mathematical proof? Can computer assisted provers output a proof? Taking a naturalized world account, I will assess the relationship between mathematics, the physical world and consciousness by introducing a significant conceptual distinction between proving and proof. I will propose that proving is a phenomenological conscious experience. This experience involves a combination of what Kurt Gödel called intuition, and what Husserl called intentionality. In contrast, proof is a function of that process — the mathematical phenomenon — that objectively self-presents a property in the world, and that results from a spatiotemporal unity being subject to the exact laws of nature. In this essay, I apply phenomenology to mathematical proving as a performance of consciousness, that is, a lived experience expressed and formalized in language, in which there is the possibility of formulating intersubjectively shareable meanings. (shrink)

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; the (...) law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the 'relay runner model,' according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the 'Baconian sciences,' i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not 'only a language' and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

Hao Wang was born in 1921 in north-east China of a westernizing father; he came to study and teach at Harvard in 1949, and he remained in the West, always feeling culturally of two distinct worlds. His deep interest in Gödel as thinker and person indicates a Confucian influence, which holds the teacher to be the great authority in both administrative as well as family matters. Wang more than once combined scientific explication with biography, since in China a student must (...) always show special respect and veneration for his teacher as a kind of father. (shrink)

A listing is given of the published writings of the logician and philosopher Hao Wang , which includes all items known to the authors, including writings in Chinese and translations into other languages.

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, Godel’s Proof by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy (...) the chance to satisfy their intellectual curiosity. (shrink)

In 1928 Edmund Husserl wrote that “The ideal of the future is essentially that of phenomenologically based (“philosophical”) sciences, in unitary relation to an absolute theory of monads” (“Phenomenology”, Encyclopedia Britannica draft) There are references to phenomenological monadology in various writings of Husserl. Kurt Gödel began to study Husserl’s work in 1959. On the basis of his later discussions with Gödel, Hao Wang tells us that “Gödel’s own main aim in philosophy was to develop metaphysics—specifically, something like the monadology of (...) Leibniz transformed into exact theory—with the help of phenomenology.” (A Logical Journey: From Gödel to Philosophy, p. 166) In the Cartesian Meditations and other works Husserl identifies ‘monads’ (in his sense) with ‘transcendental egos in their full concreteness’. In this paper I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel and to aspects of Leibniz’s original monadology. (shrink)

Godel began to seriously study Husserl's phenomenology in 1959, and the Godel Nachlass is known to contain many notes on Husserl. In this paper I describe what is presently known about Godel's interest in phenomenology. Among other things, it appears that the 1963 supplement to "What is Cantor's Continuum Hypothesis?", which contains Godel's famous views on mathematical intuition, may have been influenced by Husserl. I then show how Godel's views on mathematical intuition and objectivity can be readily interpreted in a (...) phenomenological theory of intuition and mathematical knowledge. (shrink)